The discrete fourier transform (DFT) and the discrete cosine transform (DCT) are mathematical algorithms that break down signal data or images into their sine wave components. The sine wave components are also known as the basis components or the basis functions of the transform. When pieced back together, these components reproduce the original signal or image. Heretofore known embodiments of signal or image compression systems often use the DFT or DCT to break signals or images into their sinusoidal components and save only the largest components to compress the data. The rest of the components are discarded. This can result in a significant loss of information since the signal or image's information is spread over all of its sinusoidal components. For example, the sharp details of an X-ray image are smoothed when the image is compressed at high levels, thus reducing the value of the image. If a less-lossy, lower level of compression is used, data storage requirements are increased and a greater amount of time is required for data transmission. Similar problems occur in other data compression applications.
The standard wavelet transform (SWT) was recently developed as an improvement over the DFT to better represent signals having sharp changes or discontinuities. The basis functions of the SWT are wavelets, which are functions developed by mathematicians to better match signal discontinuities. However, wavelet functions do not produce good representations of sinusoidal signals. Therefore, neither the DFT nor the SWT produces a good signal decomposition for signals consisting of both sinusoidal sections and discontinuities.
A concentrated representation of an arbitrary signal tends to transform most of the signal information into a small number of basis elements. For example, if a sinusoid is transformed, the representation might consist of one large non-zero coefficient corresponding to a sinusoidal base element of the same frequency as the signal. If the transform has the ability to produce a concentrated representation, it is possible to compress the signal to a high degree and also produce an excellent set of features for pattern matching. This occurs because since the information in the signal is concentrated into a small number of coefficients making signal features such as edges, spikes, peaks, sinusoids, and noise easily identifiable.
Neither the SWT nor the DFT produces concentrated representations for arbitrary signals because both transforms use the same set of basis functions for every signal. The DFT is best able to represent sinusoidal functions, while the SWT is best able to represent discontinuities. However, neither transform produces concentrated representations for signals whose time-frequency characteristics are not well matched to the time-frequency characteristics of its basis functions.
In most signal processing applications, it is not possible to select the transform that produces the best level of compression for a particular signal. Techniques have been developed that attempt to adapt their basis functions to a particular signal's characteristics in order to produce high levels of compression for both signal discontinuities and sinusoids. The Karhunen-Loeve Transform and the entropy-based method of Coifman and Wickerhauser are examples of such transforms. However, these types of adaptable transform techniques have the following disadvantages: 1) the amount of data required to describe the coefficients being used is so great that the advantages of adaptive transform techniques are greatly negated; and 2) they are computationally intensive and there is no reliable way to perform the techniques in an efficient manner.